On Accuracy

Mark Pottenger

I just sent the final master disks for the 1988 release of
the CCRS Horoscope Program to Astrolabe. This article is an adaptation
of one of the appendices in the manual which I thought might be of interest.

This release extends the ephemerides to the years 4,000 B.C.
through 2,500 A.D., adds timed transits and dated progressions, and puts back
planetary returns after a 5-year absence. The accuracy of all planetary
calculations has been increased, and none of these changes would have made
sense without that improvement. The rest of this article is devoted to
qualifying and clarifying that claim.

Delta t

One of the reasons any claims for accuracy in ancient dates
are subject to question is a fundamental problem with time. All theories of
planetary motion work with some form of time variable based on a consistent,
uniform and nonvarying system of time measurement. Our ordinary day of solar
(or sidereal) time based on the rotation of the earth is not such a system.
The earth’s rotation is gradually slowing on a long time-scale and also shows
small and irregular changes on a short time-scale. Delta t, meaning change or
difference in time, is a correction (fudge factor) to get from clock times tied
to the rotation of the earth to the uniform ephemeris time or dynamical time
used in planetary theories. Delta t is determined empirically by comparing
planetary and lunar theory to observations, so values for delta t are always a
year or two behind the present. The standard Astronomical Almanac gives
estimated values only a year or two into the future.

In the CCRS Horoscope Program, values for delta t
from 1620 to the present are in a disk file (DELTAT) which the program reads.
For future dates, the program just uses the last value of delta t in the file.
For dates before 1620, the program uses a formula to estimate delta t. It is
in estimating ancient values for delta t that we run into a major uncertainty.
In versions up through 1987, I used a formula in Meeus’ calculator book. In
the 1988 version I have switched to a formula in Bretagnon-Simon. The table
below shows the effect of this change.

DELTA T ESTIMATES (IN SECONDS & HOURS)

YEAR

PREVIOUS S

PREVIOUS H

CURRENT S

CURRENT H

-4000

100021

27.78

109692

30.47

-3900

96589

26.83

105948

29.43

-3800

93217

25.89

102269

28.41

-3700

89904

24.97

98655

27.40

-3600

86652

24.07

95106

26.42

-3500

83459

23.18

91622

25.45

-3400

80327

22.31

88203

24.50

-3300

77254

21.46

84849

23.57

-3200

74242

20.62

81560

22.66

-3100

71289

19.80

78336

21.76

-3000

68396

19.00

75177

20.88

-2900

65563

18.21

72083

20.02

-2800

62790

17.44

69054

19.18

-2700

60076

16.69

66090

18.36

-2600

57423

15.95

63191

17.55

-2500

54830

15.23

60357

16.77

-2400

52296

14.53

57588

16.00

-2300

49823

13.84

54884

15.25

-2200

47409

13.17

52245

14.51

-2100

45055

12.52

49671

13.80

-2000

42761

11.88

47162

13.10

-1900

40527

11.26

44718

12.42

-1800

38353

10.65

42339

11.76

-1700

36239

10.07

40025

11.12

-1600

34185

9.50

37776

10.49

-1500

32190

8.94

35592

9.89

-1400

30256

8.40

33473

9.30

-1300

28381

7.88

31419

8.73

-1200

26567

7.38

29430

8.18

-1100

24812

6.89

27506

7.64

-1000

23117

6.42

25647

7.12

-900

21482

5.97

23853

6.63

-800

19907

5.53

22124

6.15

-700

18392

5.11

20460

5.68

-600

16937

4.70

18861

5.24

-500

15541

4.32

17327

4.81

-400

14206

3.95

15858

4.41

-300

12930

3.59

14454

4.02

-200

11715

3.25

13115

3.64

-100

10559

2.93

11841

3.29

0

9463

2.63

10632

2.95

100

8427

2.34

9488

2.64

200

7451

2.07

8409

2.34

300

6535

1.82

7395

2.05

400

5679

1.58

6446

1.79

500

4883

1.36

5562

1.55

600

4146

1.15

4743

1.32

700

3470

0.96

3989

1.11

800

2853

0.79

3300

0.92

900

2297

0.64

2676

0.74

1000

1800

0.50

2117

0.59

1100

1363

0.38

1623

0.45

1200

986

0.27

1194

0.33

1300

669

0.19

830

0.23

1400

412

0.11

531

0.15

1500

215

0.06

297

0.08

1600

77

0.02

128

0.04

As you can see, the two formulae for estimating delta t
differ by hours for ancient dates (almost 3 hours at the new earliest date, and
over 1/2 hour at the previous earliest date). I switched because I hope the newer
formula is a more accurate estimate, but I don’t know how close either is to “reality”.
There have also been questions over the years about the validity of delta t,
though they mostly seem to have been settled.

Also, you can see from the table that delta t is literally
changing the time for which you are calculating planets positions by hours for
ancient dates (more than one full day before -3300).

Ephemerides

All ephemerides and theories of planetary motion involve
some attempt to fit the predictions of gravitational theories to a real world
of observations. This is essentially a statistical process—compare your
numbers to observed planetary positions, tweak your numbers or your theory,
compare again, and keep tweaking until you get the best match you can between
your theory and the observations you are using to test it. This process has
given us planetary theories which fit modern observations very well—well enough
to guide space probes to the planets and their moons. For ancient
observations, we have more uncertainty, and the delta t question. But it is
important to remember in any discussion of long-term planetary calculations, especially
for Neptune and Pluto, that the theories we are using are primarily elaborate
curve-fitting to observations in this century and some in the last century.

For several years up through 1983, the standard ephemerides
used internationally by the astronomical community were based on Sun, Mercury,
Venus and Mars positions from formulae in Volume 6 of the Astronomical
Papers, outer planets positions from Volume 12 of the Astronomical
Papers, and Moon positions from formulae in the Improved Lunar Ephemeris.

The formulae I have used in versions of the CCRS
Horoscope Program through 1987 were only intended to get positions good to
the nearest minute of arc, with Mars occasionally off by as much as two minutes.
For the 1988 version, I have upgraded to formulae which should be good to at
worst a few seconds of arc. I am using full Volume 6 formulae for the
Sun, Mercury, Venus and Mars, Volume 22 formulae for the outer planets,
and all longitude terms of one tenth of a second of arc or larger from the Improved
Lunar Ephemeris formulae for the Moon. Volume 22 is the same kind
of formula as Volume 12, but it was published about 15 years later and
has better values for Pluto. (I was using Volume 22 in the earlier
versions, but had an accuracy problem for ancient dates from having used an
approximate precession formula instead of an exact one.)

The standard ephemerides used internationally by the
astronomical community were changed in 1984 to use positions calculated at JPL
(Jet Propulsion Laboratory) with a numerical integration including the Sun,
Moon, planets, five asteroids, and effects of lunar libration and relativity.

Positions from the CCRS Horoscope Program match
pre-1984 ephemerides within less than one second of arc. Most positions are
within two-tenths of a second, with Sun positions mostly within a few
hundredths of a second. Many Moon longitudes are within one second of arc, with
almost all the rest within two seconds. The Moon’s latitude and distance can
be off quite a few seconds since I’m not using as many terms. (I plan to do
more work with the Moon formula for next year’s release.) Since the
astronomical standards changed in 1984, the match to ephemerides from 1984 on
is not as close. Many positions are still within a second of arc, most are within
two seconds, but I’ve noticed differences up to eight seconds with Neptune. (Note: this is a reduction in rated accuracy that does not reduce the internal
consistency of the positions used—see the section on returns below.)

The new astronomical standard ephemerides were developed
primarily by fitting to observations in this century, with some observations
from last century. Many old observations couldn’t be fitted to the theory and
weren’t used. For longer term ephemerides, I compared against two other
sources: charts calculated at Astro Computing Services and positions from the
program in the book Planetary Programs and Tables from -4000 to +2800 by
Bretagnon and Simon. This program gives positions for the Sun, Mercury, Venus,
Mars, Jupiter and Saturn for the years -4000 to +2800, and Uranus and Neptune for the years +1600 to +2800. All positions from Bretagnon-Simon are stated to be
accurate to within 0.01 degrees (36 seconds).

The following tables show results of a spot-check against
ACS and against Bretagnon-Simon:

CCRS vs. ACS differences in seconds of arc

YEAR

Sun

Moon

Mer

Ven

Mar

Jup

Sat

Ura

Nep

Plu

-4000

0

15

0

0

1

0

0

0

0

0

-3000

0

12

1

0

0

0

0

0

0

0

-2000

0

4

0

0

1

0

0

0

0

0

-1000

0

1

0

0

1

0

0

0

0

0

0

0

2

0

0

0

0

0

0

0

1

1000

0

0

0

0

0

0

0

0

0

0

1600

0

0

0

1

0

0

0

0

0

0

1700

0

0

1

0

1

0

0

0

0

0

1800

1

0

0

0

0

0

0

0

0

0

1900

0

1

0

0

0

0

0

0

0

0

2000

0

0

0

0

0

0

0

0

0

0

2100

0

0

0

0

1

0

0

0

0

0

2500

0

1

1

1

0

0

0

0

0

0

CCRS vs. Bretagnon-Simon differences in minutes and seconds
of arc

YEAR

Sun

Mer

Ven

Mar

Jup

Sat

Ura

Nep

-4000

6’47”

7’ 9”

7’ 4”

3’54”

1’19”

1’14”

-3000

3’52

4’ 5

3’35

4’13

2’ 8

1’46

-2000

2’ 8

2’20

2’14

1’51

1’33

1’34

-1000

58”

1’ 8”

1’ 1”

54”

52”

1’ 6”

0

27

36

34

37

33

35

1000

10

12

14

28

12

15

1600

1

9

2

4 “

5

3

6”

3

1700

2

1

1

2

3

1

1

8

1800

2

0

0

3

2

1

1

3

1900

1

2

1

1

1

1

1

2

2000

4

3

3

1

1

3

5

11

2100

10

14

14

0

2

5

3

31

2500

1’12

1’44

20

52

2

11

17

2’ 9

These tables show that CCRS positions are quite good in
comparing to the astrological standard for excellence (ACS). The comparison to
Bretagnon-Simon shows that there is room for work in comparison to recent
astronomical work. The Neptune column is especially suspicious, and I will be
looking into it for possible improvement in a future release. The entries for
2000, 2100 and 2500 clearly show the effect of delta t. My method and that
used at ACS is to use the last confirmed value for delta t into the future, but
Bretagnon-Simon uses the estimate formula for the future as well as the past.
This gets a delta t of 1,532 seconds by 2500, compared to 56 seconds from the
latest table entry—a difference of 25 minutes.

Considering the accuracy of available birth data, I believe
the planetary positions calculated by CCRS ‘88 are good enough for astrological
use.

Return times

The time of a solar, lunar, or planetary return is very
dependent on the accuracy of the basic longitudes calculated.

The Sun moves an average of 59 minutes of arc per day (with
only small variations between slowest and fastest), or 2 1/2 minutes of arc per
hour, or 2 1/2 seconds of arc per minute of time. A Sun calculated to plus or
minus one minute of arc accuracy would give you a solar return time with a
worst possible error of nearly an hour. A one minute of arc error in the natal
Sun is an error of 24 minutes of time. An additional one minute of arc error
in the return Sun is another error of 24 minutes of time. Errors will often be
in opposite directions and of different magnitudes, partially canceling each other,
but if both errors add together you get a total error of 48 minutes of time in
your return. A Sun good to plus or minus one second of arc gives you return
times with a worst possible error of 48 seconds of time. The new Sun routine
used in the CCRS Horoscope Program is accurate to a couple seconds of
arc when compared to the current astronomical standard, giving worst possible
time errors of 1 1/2 minutes. However, it is internally consistent to a few
hundredths of a second of arc, which I believe gives a better picture of return
time accuracy with a worst possible time error of 2 seconds. I mention consistency
here for a reason: if your calculations are internally consistent to a higher
precision than the accuracy of their match to observations, I think they give
return times accurate at the level of their consistency rather than at the
level of your match to observations. As an example, a watch that keeps time
correctly to a tenth of a second but is set ten minutes off gives times
consistent at the tenth of a second level, not the ten minute level. (Even if
this line of thought is full of holes, the return times are still adequate.)

A slow Moon moves under 12 degrees per day, or 30 minutes of
arc per hour, or 30 seconds of arc per minute of time. With the same logic as
described for the Sun above, we get worst possible return time errors of 4
minutes of time for a Moon good to the nearest minute of arc and 4 seconds of
time for a Moon good to the nearest second of arc. The Moon in the CCRS
Horoscope Program is good to a couple seconds of arc, giving lunar return
times within at worst 8 seconds of time.

Planetary return times are slightly more complicated because
retrogrades and velocity changes make average figures much less meaningful.
The uncertainty figure printed on the tabular pages of planetary returns is
derived from the natal and return velocities. Minutes in a day (1440) is
divided by seconds in a degree (3600), and the result is divided by the
planetary velocity in degrees per day. The answer is possible error in minutes
of time per second of arc. This figure is calculated for the natal velocity
and for the return velocity and the results are added together and multiplied
by 2 for the two second of arc accuracy level of the planetary calculations.
The answer, a worst possible error in return time, is printed as the
uncertainty figure. From this, you can see that the slower a planet is moving,
the larger your uncertainty in time. Outer planet return times are inherently
less certain than inner planet or solar and lunar returns. Return times of
planets at stations or retrograde are less certain than for direct planets.

The following table illustrates the dependency of possible
error on velocity:

ERROR IN TRANSITING TIME FOR 1 SECOND OF ARC ERROR IN
LONGITUDE

VELOCTY

ERROR

VELOCTY

ERROR

10.000

0:02

0.100

4:00

9.000

0:03

0.090

4:27

8.000

0:03

0.080

5:00

7.000

0:03

0.070

5:43

6.000

0:04

0.060

6:40

5.000

0:05

0.050

8:00

4.000

0:06

0.040

10:00

3.000

0:08

0.030

13:20

2.000

0:12

0.020

20:00

1.000

0:24

0.010

40:00

0.900

0:27

0.009

44:27

0.800

0:30

0.008

50:00

0.700

0:34

0.007

57:09

0.600

0:40

0.006

66:40

0.500

0:48

0.005

80:00

0.400

1:00

0.004

100:00

0.300

1:20

0.003

133:20

0.200

2:00

0.002

200:00

0.100

4:00

0.001

400:00

This shows time error in minutes and seconds for a range of
velocities in degrees per day. The table gives figures for a one second of arc
error. To use it to judge the accuracy of times from any program, multiply the
times by the rated accuracy (or real accuracy determined from experience if the
rated accuracy is unrealistic) of the planetary positions involved (2 for CCRS ’88).
For returns, possible error is always the sum of natal error and transiting error,
based on the velocities at each time. Always remember that this figure is the maximum
possible error and would only occur in the worst-case situation of both
natal and transiting longitudes having the maximum error and both errors adding
together instead of canceling.

An experiment for the curious: request the same planetary
return several times, giving dates from a couple days before to a couple days
after the date of the return. Look at the return times that come out. They
will be close to each other, but not the same. This effect comes from the way
returns are calculated. The program calculates the planet for the date you
enter, then corrects the date and time to get closer to the natal position. It
keeps correcting the return date and time until it gets a return position
within 0.0001 degrees (0.36 seconds of arc) of the natal position. With slow
moving planets, 0.36 seconds of arc can represent considerable time, and the
exact moment the program finds depends on where it started from.

All of this is my way of saying that planetary return times
and cusps (from anybody’s program) need to be viewed with a little healthy
skepticism.

Transiting times and progressed dates

The same kind of uncertainty discussed above for returns
also applies to any transiting times or progressed dates from any program. In
this, internal consistency of calculations doesn’t make any difference—accuracy
is the only measure to use.

For transiting times, the above formula for calculating
uncertainty of return times only needs to be changed to say that you use the
velocities of two different planets instead of two velocities for the same
planet. You still get two possible error figures from velocity and rated
accuracy and add them together for the total uncertainty.

For secondary (day for a year) progressions, apply the same
logic, then convert total uncertainty from time to date by multiplying by the
day-for-a-year ratio:

1 day : 1 year

24 hours : 365 days (approximately)

2 hours : 1 month (approximately)

1 hour : 15 days (approximately)

4 minutes : 1 day (approximately)

For every four minutes of time uncertainty in the planetary
calculations involved, you have about one day of uncertainty in the dates of
secondary progressions. (All of these ratios except the first are
approximate. The actual ratio used in the CCURRENT module of the CCRS
Horoscope Program is one day to one year of 365.24219879 days for tropical
charts or 365.25636042 days for sidereal or precessed charts.)

Details

The geocentric positions used in the CCRS Horoscope
Program are apparent positions for the true equinox of date. This means
they are positions as you would have seen things if you had gone out and looked
at the sky at that date and time. In contrast, the heliocentric positions and
the positions shown from the perspectives of other planets are all geometric
positions for the mean equinox of date. Geometric positions are where the
planets are at the moment, rather than where we see them. The difference
between geometric and apparent positions is the motion of the planet in the
time it takes light to get from the planet to the viewer. It is called
aberration or the light-time correction. The difference between the mean
equinox and the true equinox is called nutation. At the moment, the CCRS
program corrects for nutation in longitude, but does not correct for nutation
in obliquity when deriving right ascension and declination from longitude and
latitude.

Another kind of correction to positions that some people use
is parallax. A parallax correction allows for the difference in perspective
between the center of the earth and the actual observer’s position on the
surface of the earth. The CCRS Horoscope Program doesn’t currently
offer a parallax correction option, though it is under consideration for a
future release.

A concept related to parallax is whether we give positions
for the centers or the surfaces of the Sun, Moon and planets. All positions
normally given are for centers, but the Sun and Moon are so large that their
surfaces extend about 15 minutes of arc in each direction. People dealing with
conjunctions, rise & set times, and ingresses of these two bodies need to
be conscious of this extended size.

New and old

The new disk ephemerides are used everywhere in the
program. The new Sun, Moon, Mercury, Venus, and Mars routines are used in
calculating all permanently stored charts, timed progressions and transits, and
creating or printing ephemerides. The same inner planet and Moon routines as
last year are used in quick on-screen charts, untimed charts, and dynamic
astrology graphs. The reasons for keeping the older routines in these places
is to keep run times reasonable (the new routines are considerably slower than the
old routines) and because these sections don’t need second of arc accuracy.

Standards and thanks

I view Neil Michelsen’s Astro Computing Services as a
standard for accuracy in the astrological community. I want to say a special
thank you to Neil for a listing letting me match his dynamic astrology
routines, planetary masses letting me get a barycenter position consistent with
his, and a listing that let me find why my Moon positions were differing from
his more in the past.

I have used values from other astrological sources for the
origin and orientation of the invariable plane and the longitude and latitude
of the galactic center, but I will revise them if I find more recent values in
an astronomical source.